A tea party is arranged for 16 persons along two sides of a long table with 8 chairs on each side.
Question:

A tea party is arranged for 16 persons along two sides of a long table with 8 chairs on each side. Four persons wish to sit on one particular side and two on the other side. In how many ways can they be seated?

Solution:

A tea party is arranged for 16 people along two sides of a long table with 8 chairs on each side.

4 people wish to sit on side AA (say) and two on side BB (say).

Now, 10 people are left, out of which 4 people can be selected for side AA in 10C4 ways.

And, from the remaining people, 6 people can be selected for side B in 6C6 ways.

$\therefore$ Number of selections $={ }^{10} C_{4} \times{ }^{6} C_{6}$

Now, 8 people on each side can be arranged in $8 !$ ways.

$\therefore$ Total number ways in which the people can be seated $={ }^{10} C_{4} \times{ }^{6} C_{6} \times 8 ! \times 8 !=10_{C_{4}} \times(8 !)^{2}$

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